Abstract: We introduce two values for cooperative games with communication graph structure. In cooperative games without restrictions on cooperation the classical Shapley value distributes the worth of the grand coalition among the players by taking into account the worths that can be obtained by any coalition of players, however it does not take into account the role of the players when communication between the players is restricted. Existing values for communication graph games such as the Myerson value and the average tree solution only consider the worths of connected coalitions and only in this way they respect the communication restrictions. They do not take into account the position of a player in the graph in the sense that in the unanimity game on the grand coalition all players are treated equally when the graph is connected, and so, the players with a more central position in the graph get the same payoff, as players which are not central. The two new values take into account the position of a player in the graph. The first one respects centrality, but not the communication ability of a player. The second value respects both centrality and the communication ability of each player, which implies that in unanimity games players that do not generate worth but are needed to connect worth generating players are treated similar to the latter players, and simultaneously players that are more central in the graph get bigger shares than players which are less central. For both newly introduced values we provide axiomatic characterization on the class of connected cycle-free graph games.